Assignment 1

(100)

You can use the Geogebra Program to
do
this assignment. Click
here to download it. Use the Download button unless you
already have java installed on your computer. You should use the
template here to enter your answers in. Just paste your images
from Geogebra into the boxes.

You can also just use a compass and
straightedge. You will
probably want to copy your work at each stage.

Part
I: Triangles.

Draw a Vesica
Pisces.

Connect the two
points where the circles overlap with a line. Connect the two centers
of the circles with a line as well

Complete the two
equilateral triangles formed by the points. Extend the sides of the top
triangle down to meet the circles.

Complete the two
new triangles formed by the point of intersection of these extended
lines with the circles. You can erase all the lines except the large
triangle now.

Connect each
midpoint with the vertex across from it.Where have you seen this triangle figure before? It
may help to erase the circles.

Part
II. Squares.

Start with a
Vesica Pisces and draw a horizontal line connecting the midpoints. Then
draw a line perpendicular to the line at its two endpoints.
1a. ( Compass and penicl
users only): you will have to do the following
construction ot get the prependicular lines.:Draw a circle with
center B and radius BC. Draw another circle with center B and radius
BD. Draw a circle with center D and radius DB. Draw a circles with
center C and radius CB and CA. We will be interested in the
points where these
overlap in the center. You need not finish the circles to the far right
and left. This seems complex, but you are reallyjust
constructing perpendicular lines at points B and C.BC will form
the base of the square.

Mark
the points where the perpendicular lines meet the two circle at the top
and connect the four points to complete the square. you can hide the
perpendicular lines.

3.
Mark the midpoints if the
bottom and top of the square, using the center line of the vesica
pisces. You
may now erase everything but the square. Save at this point, or make a copy. You will need this again later.

Find the midpoints of the two sides of the square. You
will have to use
the line bisector in Geogebra. If doing it by hand you can measure to
find an
approximate midpoint, or, better, use these
instructions to bisect the lines manually.

Now
connect the diagonal points of
the square. Make a smaller square by connecting the fourmidpoints of the lines
that make the original
square.Now connect
the diagonals of
this smaller square. Where have you seen this diagram before?

4.
Make a smaller square inside
the second square by connecting its midpoints (marked by the diagonals
of the
original square) in the same manner as above.Now make a fourth larger square outside the original
square. Extend the
two midpoint lines of the original square, and construct a line at the
top left
corner that is parallel to the diagonal until that line meets the extended midpoint lines. Repeat for the other
three
corners.

How is each square related
to the diagonal of
the next smaller square? What are the relationships betweenthe sizes of the four
squares?

5.

Start with a square
again, from step 3 above. Draw the diagonal from point A to D. Draw the
lines perpendicular to that diagonal AD at points A and D. Then draw a
Vesica Pisces with a circle center A, radius AD and a circle center D,
radius DA.

5a..

Mark
where the circles intersect the perpendicular lines and complete the
square built on the diagonal AD, by making the segments DF, FE, and EA.

6. Now make
another diagonal CB in the original square. and repeat all the
steps from number 5 for this diagonal to create another square based
upon it, CBHG.

How many squares do you see?

Indicate the relationship between the areas and the sides of the different size squares.

Square

area

side

Unit square ABDC

1

1

larger square

smaller square

III.The Golden Section and the Pentagon:

1. Construct a Golden rectangle.

Start with a square. Use the regular
polygon tool
or start from a copy of the square constructed above.

Bisect the bottom of the square and
then continue that
bottom segment in both directions.

Draw a circle with center E and
radius ED to intersect the
bottom line. You are inscribing the square in a semi-circle. Mark the
point
where the circle hits the line F. The line AF is cut by B in the golden
section.

Mark the other point where the circle
hits the bottom line
G. Extend CD in both directions. Raise perpendiculars up at F and G.Mark the two points where
these hit line
CD,H and I. GHIF
is a Square Root of 5
rectangle and ACIF and GHDB are Golden Rectangles.

2. The Golden Spiral.

Start with a Golden rectangle ACIF
above. Note that BDIF is
also a golden rectangle.

Measure out on DB and IFa length equal to DI. Mark these points J and K. Make the
square JDIK

BJKF is also a Golden Rectangle.

Repeat the procedure a couple more
times.

Draw an arc from A to D with radius
BA. Then do the same
with the next smaller square: an arc from D to K with radius JD.

Continue with the next smaller square
and so on as far down
as you can get. This is the Logarithmic or golden spiral.

3. Pentagon:

Start with a line
divided in a Golden section, such as ABF from above. You can also
reconstruct
one using the square root of 5 rectangle method from above.

Draw circle with center A and radius
AB and another circle
with center B and radius BA.

Now draw a circle with center A and
radius AF. Then another
circle with center B with the same radius AF. (You will have to measure
AF and
use the circle with determined radius function in Geogebra) Mark the
points where
the two large circles intersect each other and the two small circles.
Connect
each of these points with each other and with AB to make the pentagon.

4. Pentagram Star.

Start with the pentagon. You may
erase all the guidelines.
Connect each vertex with the one directly opposite it. This will give
you a
pentagram star inside the pentagon.

You can repeat this process again
within the internal
pentagon.
Extend each of the sides of the original pentagon to make a larger
pentagram
outside.

How many instances of the golden
relationship can you find
between the parts of the pentagram?

IV.
The Platonic Solids:

Start with a vesica pisces divided
into 4 triangles asin
part I above:

Remove the circles and fold you have a tetrahedron.

Extend the vesica pisces to six
circles and use it to trace
out these 6 squares:

The same pattern with 5 circles will
give the octahedron:

Use thesame pattern
with 8 circles for the icosahedron:

Just hand in the drawn or printed
templates.

Extra Credit ONLY

You can cut them out and actually
construct the solids for
extra credit. If you like , you can print out these already drawn
templates to
make your models.

PART V: Finding
the Golden Section

There
are a number of ways you can do the measurements in this section: (the
three kinds of measurements below in questions 1, 2, and 3.)1.
Use a ruler and calclulate the ratio of sides to see how close it comes
to .618 or 1.618. This will be the least exact method.2. Print out these templates: vertical and horizontal in different sizes or use them on the screen and use them to measure objects.Here is another template.3. Use one of these programs.
Both of these are shareware. You can use then free for 30 days.They produce a template that appears transparently over your screen you can move around and resize to see if things are in the golden proportion.Phi Matrix is simpler to use. Atrise has more features.Phi Matrix http://www.phimatrix.com/PhiMatrix.exe

Required Measurements.
You must do the minimum. You can get extra
points by doing more and finding really interesting things. You will
get less credit if you go to a site that already has collected images
in the golden section for you. Try to find your own.

1. Measure at least 5 commonly used man made objects. Can you
find at least one that has proportions in the golden section?

Object

Feature

Golden Section?

2. Look at some pictures of
art works or architecture. Find at least 5 that have prominent features
in the golden section.List their names and the features
measured. You might also find this template
useful in looking at larger objects. You may find this table helpful in
reporting your results

Name

Feature

Artist (if known)

Web address (if from web) or source

Image
(optional)

3. Look at some natural
objects and
parts of natural objects (any living thing, part of a living thing, or
highly organized inorganic object, such as a crystal. Don't use
anything that has been reshaped by Man, such a cut gem or carved wood.)
Find and list at least 8 examples (at least 4 should be from different
objects, not parts of the same object) of the golden section. Did you
find any organized natural objects whose main divisions were not in the
golden proportion? List them. You may find this table helpful in
reporting your results